Relating the total domination number and the annihilation number for quasi-trees and some composite graphs

The total domination number \(\gamma_{t}(G)\) of a graph \(G\) is the cardinality of a smallest set \(D\subseteq V(G)\) such that each vertex of \(G\) has a neighbor in \(D\). The annihilation number \(a(G)\) of \(G\) is the largest integer \(k\) such that there exist \(k\) different vertices in \(G...

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Bibliographic Details
Published inarXiv.org
Main Authors Hua, Hongbo, Hua, Xinying, Klavžar, Sandi, Xu, Kexiang
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 23.04.2022
Subjects
Apexes
Graphs
Trees (mathematics)
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Summary:The total domination number \(\gamma_{t}(G)\) of a graph \(G\) is the cardinality of a smallest set \(D\subseteq V(G)\) such that each vertex of \(G\) has a neighbor in \(D\). The annihilation number \(a(G)\) of \(G\) is the largest integer \(k\) such that there exist \(k\) different vertices in \(G\) with the degree sum at most \(m(G)\). It is conjectured that \(\gamma_{t}(G)\leq a(G)+1\) holds for every nontrivial connected graph \(G\). The conjecture has been proved for graphs with minimum degree at least \(3\), trees, certain tree-like graphs, block graphs, and cactus graphs. In the main result of this paper it is proved that the conjecture holds for quasi-trees. The conjecture is verified also for some graph constructions including bijection graphs, Mycielskians, and the newly introduced universally-identifying graphs.
ISSN:2331-8422